Quantitative Big Imaging

With inconsistent or every changing illumination it may not be possible to apply the same threshold to every image.

Here is a bone slice

1. Find the larger cellular structures (osteocyte lacunae)
2. Find the small channels which connect them together

The first task

is easy using a threshold and size criteria (we know how big the cells should be)

The second

is much more difficult because the small channels having radii on the same order of the pixel size are obscured by partial volume effects and noise.

• The cortex is barely visible to the human eye
• Tiny structures hint at where cortex is located

• A simple threshold is insufficient to finding the cortical structures
• Other filtering techniques are unlikely to magicially fix this problem

Given that applying a threshold is such a common and signficant step, there have been many tools developed to automatically (unsupervised) perform it. A particularly important step in setups where images are rarely consistent such as outdoor imaging which has varying lighting (sun, clouds). The methods are based on several basic principles.

Histogram-based methods

Just like we visually inspect a histogram an algorithm can examine the histogram and find local minimums between two peaks, maximum / minimum entropy and other factors

• Otsu, Isodata, Intermodes, etc

Image based Methods

These look at the statistics of the thresheld image themselves (like entropy) to estimate the threshold

Results-based Methods

These search for a threshold which delivers the desired results in the final objects. For example if you know you have an image of cells and each cell is between 200-10000 pixels the algorithm runs thresholds until the objects are of the desired size

• More specific requirements need to be implemented manually

For some images a single threshold does not work

• large structures are very clearly defined
• smaller structures are difficult to differentiated (see partial volume effect)

ImageJ Source

Now we apply two important steps. The first is to remove the objects which are not cells (too small) using an opening operation.

The second step to keep the between pixels which are connected (by looking again at a neighborhood \( \mathcal{N} \)) to the air voxels and ignore the other ones. This goes back to our original supposition that the smaller structures are connected to the larger structures

As we briefly covered last time, many measurement techniques produce quite rich data.

• Digital cameras produce 3 channels of color for each pixel (rather than just one intensity)
• MRI produces dozens of pieces of information for every voxel which are used when examining different contrasts in the system.
• Raman-shift imaging produces an entire spectrum for each pixel
• Coherent diffraction techniques produce 2- (sometimes 3) diffraction patterns for each point. \[ I(x,y) = \hat{f}(x,y) \]

We give as an initial parameter the number of groups we want to find and possible a criteria for removing groups that are too similar 1. Randomly create center points (groups) in vector space 1. Assigns group to data point by the “closest” center 1. Recalculate centers from mean point in each group 1. Go back to step 2 until the groups stop changing

What vector space to we have?

• Sometimes represent physical locations (classify swiss people into cities)
• Can include intensity or color (K-means can be used as a thresholding technique when you give it image intensity as the vector and tell it to find two or more groups)
• Can also include orientation, shape, or in extreme cases full spectra (chemically sensitive imaging)

Note: If you look for 2 groups you will almost always find two groups, whether or not they make any sense

objColor=kmeans(indata,2);

Or just the orientation

objColor=kmeans(indata(:,[aisoCol,thCol]),2);

Fuzzy classification based on Fuzzy logic and Fuzzy set theory and is a general catagory for multi-value logic instead of simply true and false and can be used to build IF and THEN statements from our probabilistic models.

Instead of

\[ P(\{\vec{x} , I(\vec{x})\} | \alpha) \propto P(\alpha) + P(f(\vec{x}) | \alpha)+ \] \[ P(\sum_{x^{\prime} \in \mathcal{N}} f(\vec{x^{\prime}}) | \alpha) \]

Clear simple rules

which encompass aspects of filtering, thresholding, and morphological operations

• IF the intensity if dark (\( <100 \))
  • AND a majority of the neighborhood (\( \mathcal{N} \)) values are dark (\( <100 \))
• THEN it is a cell

Once we have a clearly segmented image, it is often helpful to identify the sub-components of this image. The easist method for identifying these subcomponents is called component labeling which again uses the neighborhood \( \mathcal{N} \) as a criterion for connectivity, resulting in pixels which are touching being part of the same object.

In general, the approach works well since usually when different regions are touching, they are related. It runs into issues when you have multiple regions which agglomerate together, for example a continuous pore network (1 object) or a cluster of touching cells

The more general formulation of the problem is for networks (roads, computers, social). Are the points start and finish connected?

We start out with the network and we grow Begin to its connections. In a brushfire-style algorithm

• For each point \( (x,y)\in\{\text{Begin},\text{Pixels Grown}\} \)
  • For each point \( (x^{\prime},y^{\prime})\in\mathcal{N}(x,y) \)
  • Set the label to Pixels Grown
• Repeat until no more labels have been changed

Same as for networks but the neighborhood is defined with a kernel (circular, full, line, etc) and labels must be generate for the image

• Assign a unique label to each point in the image \[ L(x,y) = y*\text{Im}_{width}+x \]

• For each point \( (x,y) \)

  • Check each point \( (x^{\prime},y^{\prime})\in\mathcal{N}(x,y) \)
  • Set \[ L(x,y)=\text{min}(L(x,y),L(x^{\prime},y^{\prime})) \] \[ L(x^{\prime},y^{\prime})=L(x,y) \]

• Repeat until no more \( L \) values are changed

The image very quickly converges and after 4 iterations the task is complete. For larger more complicated images with thousands of components this task can take longer, but there exist much more efficient algorithms for labeling components which alleviate this issue.